Not having a heartbeat and not breathing doesn’t mean you’re dead. Intensive care departments are literally full of people with medically paralysed breathing muscles (i.e. not breathing) on ventilation machines. People go onto heart/lung bypass machines everyday to have heart surgery and their heart is stopped. You just need to keep oxygenated blood going around, keeping those tissues alive till you get the heart and breathing back online (this is what CPR is trying to do).
When the brain stem is dead tissue, then you’re truly dead (but even then you can be kept “alive” artificially if you’re already on a ventilation machine in a suitable intensive care).
The medical community has long since moved on from the cardiovascular definition of death.
UpToDate.com is about the only source I can be bothered mustering up for an internet disagreement at this time of night:
Death is an irreversible, biologic event that consists of permanent cessation of the critical functions of the organism as a whole [1]. This concept allows for survival of tissues in isolation, but it requires the loss of integrated function of various organ systems. Death of the brain therefore qualifies as death, as the brain is essential for integrating critical functions of the body. The equivalence of brain death with death is largely, although not universally, accepted [2,3]. Brain death implies the permanent absence of cerebral and brainstem functions.
Also this video seems to explain what I’m trying to say, although I’m not going to watch the whole thing at this hour and I only skimmed through it: https://m.youtube.com/watch?v=5IhxRSaJ74E